That's called a shower tray, and yes, it would be a good idea I think. Although it sounds like you're going to have to think about plumbing too.
Blaze, does your bathroom floor have no downwards angle towards the plug? And the plug is not near the shower so water could potentially just sit there on the floor all day? Sounds very badly designed - or very badly pictured in my mind... I can see my kids turning the entire bathroom into a slip 'n' slide nightmare, complete with hospital visits and the like
The water-proof board I was talking about is a regulation gib-board in New Zealand for bathroom installation. I am not sure whether sheetrock is the same thing as gib-board... and I would have no idea where to go to buy it here.
I think it's a good idea to have a talk with your landlord, as he/she will probably know a person who could do it for you or where you could buy materials. At any rate, you'll need permission to alter the premises.